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Length and dimension modulo a Serre category. (English) Zbl 0878.13007
A module \(M\) over a commutative ring \(R\) is called \({\mathcal C}\)-simple with regard to a Serre category \({\mathcal C}\) if for each submodule \(N\) either \(N\in{\mathcal C}\) or \(M/N\in{\mathcal C}\). Using this notion of simplicity the author defines recursively \({\mathcal C}\)-dimension and \({\mathcal C}\)-length for a module, which is shown to generalize various concepts of dimensions, among others the one introduced by R. Rentschler and P. Gabriel for noetherian modules [C. R. Acad. Sci., Paris, Sér. A 265, 712-715 (1967; Zbl 0155.36201)]. In the case of \(M=R\) the results allow certain generalizations of the classical Krull dimension.
MSC:
13D05 Homological dimension and commutative rings
13C15 Dimension theory, depth, related commutative rings (catenary, etc.)
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[1] DOI: 10.1017/S0305004100075137 · Zbl 0757.13006 · doi:10.1017/S0305004100075137
[2] Rentschler K., C.R. Acad. Sci. Paris 265 pp 712– (1967)
[3] DOI: 10.1093/qmath/26.1.269 · Zbl 0311.13006 · doi:10.1093/qmath/26.1.269
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